An interaction sample

The interaction sample, parsed

The Lexicon

A Linear Regression Model for MLU

Model Fitting

\(MLU(child_i) \sim Normal(\alpha + \beta \cdot MLU(mother_i), \sigma)\)

mod_utt_td <- lm(formula = CHI_MLU ~ MOT_MLU, data = data)
mod_utt_td$coefficients

## (Intercept)     MOT_MLU 
##  -1.0680205   0.7813712

\[\begin{align} MLU(child) &= -1.07 + 0.78 \cdot MLU(mother)\\ R^2_{adj}&=0.32 \end{align}\]

• The intercept (-1.1) represents the number of words uttered by the child when the mother says nothing (0)
• The slope (0.78) represents the change in the number of words uttered by the child when the number of words uttered by the mother increases by one

Regression and Prediction

• Linear models allow to make predictions for any given predictor variable value (any given number of words uttered by the parent)
• The function predict() retrieves parameters of the fitted model (in our case, the intercept and the slope) to make predictions about the outcome variable, given some values of the predictor(s)
  • Second line of the linear model
• What is \(MLU_{CHILD}\), given that \(MLU_{MOT}=4.5\)?

\[\begin{align} MLU(child) &= -1.07 + 0.78 \cdot MLU(mother)\\ &= -1.07 + .78 \cdot 4.5\\ &= 2.44 \end{align}\]

Plotting the prediction

Multiple Fit and Prediction

mod_utt_both <- lm(formula = CHI_MLU ~ MOT_MLU * Diagnosis, data = data)
mod_utt_both$coefficients

##         (Intercept)             MOT_MLU         DiagnosisTD MOT_MLU:DiagnosisTD 
##          -0.4088249           0.5608042          -0.9397872           0.3198611

\[\begin{align} MLU(child_{ASD}) &= -0.41 + 0.56 \cdot MLU(mother)\\ MLU(child_{TD}) &= -.41 + .56 \cdot MLU(mother) -.94 + .32 \cdot MLU(mother)\\ &= -1.35 + .88 \cdot MLU(mother)\\ R^2_{adj}&= .35 \end{align}\]

Multiple Regression and Prediction

• What are \(MLU(child_{ASD})\) and \(MLU(child_{TD})\), given that \(MLU(mother)=4.5\)?

\[\begin{align} MLU(child_{ASD}) &= -0.41 + 0.56 \cdot MLU(mother)\\ &= -0.41 + 0.56 \cdot .45\\ &= 2.11 \end{align}\]

\[\begin{align} MLU(child_{TD}) &= -1.35 + .88 \cdot MLU(mother)\\ &= -1.35 + .88 \cdot.45\\ &= 2.61 \end{align}\]

Assessing normality of residuals

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

The general bayesian workflow (Gelman, 2013)

For each statistical problem, we follow three general steps:

1. Model Building: likelihood + priors
2. Model Updating using information contained in data and Bayes theorem (aka approximating the posterior probability)
3. Model Evaluation: fit, assumptions, results summarization, readjusting the model

The bayesian workflow in detail (Gelman, 2020)

Our first model

formula1 <- brms::bf(CHI_MLU ~ MOT_MLU * Diagnosis)
model1 <- brms::brm(formula = formula1,
                    data = data,
                    file = 'data/w5/example')
model1

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: CHI_MLU ~ MOT_MLU * Diagnosis 
##    Data: data (Number of observations: 352) 
##   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup draws = 4000
## 
## Population-Level Effects: 
##                     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept              -0.40      0.31    -1.02     0.20 1.00     1814     2230
## MOT_MLU                 0.56      0.08     0.40     0.73 1.00     1784     2338
## DiagnosisTD            -0.96      0.53    -2.01     0.08 1.00     1373     1770
## MOT_MLU:DiagnosisTD     0.32      0.13     0.06     0.58 1.00     1352     1812
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     0.75      0.03     0.70     0.81 1.00     2975     2481
## 
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).